# Theoretical Analysis of Heteroscedastic Gaussian Processes with Posterior Distributions

arXiv: https://arxiv.org/abs/2409.12622 · HTML: https://mlsanity.com/paper/2409.12622/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 41 references that are in the map.
- First posted on arXiv 19 Sep 2024.

## Builds on (most cited first, 14 of 14)

- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Safe Learning of Regions of Attraction for Uncertain, Nonlinear Systems with Gaussian Processes](https://mlsanity.com/paper/1603.04915/) (2016)
- [Non-Stationary Gaussian Process Regression with Hamiltonian Monte Carlo](https://mlsanity.com/paper/1508.04319/) (2015)
- [Risk-averse Heteroscedastic Bayesian Optimization](https://mlsanity.com/paper/2111.03637/) (2021)
- [On Controller Tuning with Time-Varying Bayesian Optimization](https://mlsanity.com/paper/2207.11120/) (2022)
- [Heteroscedastic Bayesian Optimisation for Stochastic Model Predictive Control](https://mlsanity.com/paper/2010.00202/) (2020)
- [Stochastic Optimization for Trajectory Planning with Heteroscedastic Gaussian Processes](https://mlsanity.com/paper/1907.07521/) (2019)
- [Matching models across abstraction levels with Gaussian Processes](https://mlsanity.com/paper/1605.02190/) (2016)
- [Stochastic Cell Transmission Models of Traffic Networks](https://mlsanity.com/paper/2304.11654/) (2023)
- [Gaussian Processes with State-Dependent Noise for Stochastic Control](https://mlsanity.com/paper/2305.16229/) (2023)
- [Polynomial Chaos Expanded Gaussian Process](https://mlsanity.com/paper/2405.01052/) (2024)
- [An Uncertainty-aware, Mesh-free Numerical Method for Kolmogorov PDEs](https://mlsanity.com/paper/2405.05626/) (2024)

## Abstract (arXiv)

This study introduces a novel theoretical framework for analyzing heteroscedastic Gaussian processes (HGPs) that identify unknown systems in a data-driven manner. Although HGPs effectively address the heteroscedasticity of noise in complex training datasets, calculating the exact posterior distributions of the HGPs is challenging, as these distributions are no longer multivariate normal. This study derives the exact means, variances, and cumulative distributions of the posterior distributions. Furthermore, the derived theoretical findings are applied to a chance-constrained tracking controller. After an HGP identifies an unknown disturbance in a plant system, the controller can handle chance constraints regarding the system despite the presence of the disturbance.